ISRN.APPLIED.MATHEMATICS ISRN Applied Mathematics 2090-5572 Hindawi Publishing Corporation 856396 10.1155/2014/856396 856396 Research Article A Modified Approach to the New Solutions of Generalized mKdV Equation Using (G/G)-Expansion http://orcid.org/0000-0001-8214-948X Zhou Yu 1 Wang Ying 1, 2 Djidjeli K. Mishuris G. 1 School of Mathematics and Physics Jiangsu University of Science and Technology Jiangsu 212003 China 2 Benjamin M. Statler College of Engineering and Mineral Resources West Virginia University Morgantown, WV 26505 USA wvu.edu 2014 532014 2014 18 11 2013 08 12 2013 5 3 2014 2014 Copyright © 2014 Yu Zhou and Ying Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The modified (G/G)-expansion method is applied for finding new solutions of the generalized mKdV equation. By taking an appropriate transformation, the generalized mKdV equation is solved in different cases and hyperbolic, trigonometric, and rational function solutions are obtained.

1. Introduction

The evolutions of the physical, engineering, and other systems always behave nonlinearly; hence many nonlinear evolution equations have been introduced to interpret the phenomena. Many kinds of mathematical methods have been established to investigate the solutions of those nonlinear evolution equations both numerically and asymptotically, while the exact solutions are of particular interests. In recent decades, with the rapid progress of computation methods, many effective calculating approaches have been developed, for example, the tanh-coth expansion [1, 2], F-expansion [3, 4], Painlevé expansion , Jacobi elliptic function method , Hirota bilinear transformation , Backlund/Darboux transformation [8, 9], variational method , the homogeneous balance method , exp-function expansion , and so on. However, a unified approach to obtain the complete solutions of the nonlinear evolution equations has not been revealed.

Within recent years, a new method called (G/G)-expansion  has been proposed for finding the traveling wave solutions of the nonlinear evolution equations. Many equations have been investigated and many solutions have been found using the method, including KdV equation, Hirota-Satsuma equation , coupled Boussinesq equation , generalized Bretherton equation , the mKdV equation , the Burgers-KdV equation, the Benjamin-Bona-Mahony equation , the Whitham-Broer-Kaup-like equation , the Kolmogorov-Petrovskii-Piskunov equation , KdV-Burgers equation , and Drinfeld-Sokolov-Satsuma-Hirota equation .

The mKdV equation, a modified version of the Korteweg-de Vries (KdV) equation, has been investigated extensively since Zabusky showed how this equation depict the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam (FPU) model . Afterwards, this equation has been used to describe the evolution of internal waves at the interface of two layers of equal depth . Generally, the KdV theory describes the weak nonlinearity and weak dispersion while, in the study of nonlinear optics, the complex mKdV equation has even been used to describe the propagation of optical pulses in nematic optical fibers when we go beyond the usual weakly nonlinear limit of Kerr medium . In some cases, the exponential order may be not a positive integer, but just a real number. After this kind of generalized mKdV equation  has been introduced, interests of investigating the solutions of it  have been inspired; then the standard expansion methods cannot be applied, and some kinds of transformation are needed.

In this paper, we modify the standard (G/G)-expansion method and use it to solve the generalized mKdV equation. In next section, we briefly introduce the modified (G/G)-expansion method while in Section 3, we apply it to find some types of new solutions of mKdV equation, and the last section gives the summary and conclusion.

2. An Introduction to the Modified (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mo mathvariant="bold">/</mml:mo><mml:mi>G</mml:mi></mml:math></inline-formula>)-Expansion Method

Recently, a new approach called (G/G)-expansion has been proposed dealing with the problems of finding solutions of nonlinear evolution equations  and some modifications to this method have been developed. Here we briefly outline the main steps of the modified (G/G)-expansion method in the following.

Step 1.

We consider a given (1)P(u,ut,ux,uxt,utt,uxx,)=0, where P is a polynomial for its arguments and u=u(x,t) is the unknown function. Introducing the new variable ξ and supposing that u(ξ)=u(x,t) and ξ=x-vt, hence, the partial differential equation (PDE) (1) is reduced to an ordinary differential equation (ODE) for u=u(ξ) as (2)P(u,-vu,u,v2u′′,-vu′′,u′′,)=0.

Step 2.

For ODE (2) above, the solution could be expressed by a polynomial in G/G as (3)u(ξ)=i=-mmai(GG)i, where G=G(ξ) is the solution of a second order linear ODE (4)G′′(ξ)+λG(ξ)+μG(ξ)=0 with constants λ, μ to be determined later. Positive integer m is an index yet undetermined which should be calculated by the balance between the highest order derivatives and the nonlinear terms from ODE (2). By solving (4), it is apparent that the form of G/G in three different cases read as follows.

( 1 ) When λ2-4μ>0, (5)GG=λ2-4μ2×(+C2sinhλ2-4μ2ξ)-1(C1sinhλ2-4μ2ξ+C2coshλ2-4μ2ξ)×(C1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ)-1)-λ2.

( 2 ) When λ2-4μ<0, (6)GG=4μ-λ22×(+C2sin4μ-λ22ξ)-1(-C1sin4μ-λ22ξ+C2cos4μ-λ22ξ)×(C1cos4μ-λ22ξ+C2sin4μ-λ22ξ)-1)-λ2.

( 3 ) When λ2-4μ=0, (7)GG=C2C1+C2ξ-λ2, where C1 and C2 in above three solutions (5), (6), and (7) of (4) are integrate constants.

Step 3.

Substituting solution (3) into ODE (2) using (4), we have a set of differential equations. Collecting all terms together according to the same order of G/G, the left-hand side of ODE (2) becomes a long expression in the form of polynomial of G/G. Making all coefficients of each order of G/G equal to zero, the solution sets of the parameters λ, μ, ai, and v will be got after solving the algebra equations.

Step 4.

Based on the last step, we now have the solutions of the coefficient algebra equations and after substituting the parameters ai, v, and so forth into solution (3), we could reach different types of travelling wave solutions of the PDE (1).

3. Application to the Generalized mKdV Equation

Now, we consider the generalized mKdV equation with the form  (8)ut+αuγux+βuxxx=0, while parameters α and β are real constants. Denoting the travelling wave solution as (9)u(x,t)=u(ξ), with ξ=x-vt, then the PDE (8) becomes an ODE. After integrating with single variable ξ and setting the integrate constant to zero, we have (10)-vu+α1+γu1+γ+βu′′=0. It can be easily seen that the standard (G/G)-expansion cannot be applied directly to this situation because of the arbitrary power index γ, which results in the noninteger power index of G/G. So it is necessary to introduce a transformation to deal with it via assuming that (11)u(ξ)=w1/γ(ξ); then (10) becomes (12)βγ(1+γ)ww′′+β(1-γ2)w2-vγ2(1+γ)w2+αγ2w3=0. The balancing between the highest order nonlinear term and the highest order derivative term leads to the balance parameter m=2; hence the solution (3) could be expressed as (13)w(ξ)=a0+a1ϕ(ξ)+a2ϕ2(ξ)+a-1ϕ(ξ)+a-2ϕ2(ξ), with ϕ(ξ)=G(ξ)/G(ξ) and a0, a1, a2, a-1, and a-2 being constants to be determined later.

Substituting (13) into (12), making use of (4), a polynomial of ϕ(ξ) is obtained; a set of nonlinear algebra equations about undetermined constants a0, a1, a2, a-1, a-2, λ, μ, and v are reached through setting the coefficients of each order of ϕ to zero. These equations are expressed as follows.

( 1 )   ϕ0-order: (14)-v(a02-2a1a-1-2a2a-2)γ3+{[4(a1a-1+4a2a-2)λ2+(17a1a-2+17μa2a-1+a0a-1+μa0a1)λ+(2a0a2-a12)μ2+8(a1a-1+4a2a-2)μ-a-12+2a0a-2]β+(αa03-va02+6α(a1a-1+a2a-2)a0+(3αa12-2va2)a-2+(3αa2a-12-2va1a-1)}γ2+[2(a1a-1+4a2a-2)λ2+(μa0a1+a0a-1+9μa2a-1+9a1a-2)λ+2a0a2μ2+4(a1a-1+4a2a-2)μ+2a0a-2]βγ+[-2(a1a-1+4a2a-2)λ2-8(μa2a-1+a1a-2)λ+μ2a12-4(a1a-1+4a2a-2)μ+a-12]β=0.

( 2 )   ϕ1-order: (15)-2v(a0a1+a2a-1)γ3+{β(a0a1+9a2a-1)λ2+[βμ(6a0a1-a12)+8β(a1a-1+4a2a-2)]λ-2βμ2a1a2+2β(a0a1+9a2a-1)μ+3(a12a-1+a02a1+2a1a2a-1+2a0a2a-1)α-2va0a1-2va2a-1+8βa1a-2(a0a1+9a2a-1)(6a0a1-a12)}γ2+β(γa0a1+5γa2a-1-4a2a-1)λ2+β{[(2+γ)a12+6γa0a2]μ+4(γ-1)(a1a-1+4a2a-2)[(2+γ)a12+6γa0a2]}λ+β[2(γ+2)a1a2μ2+2(γa0a1+5γa2a-1-4a2a-1)μ+4(γ-1)a1a-1a2μ2]=0.

( 3 )   ϕ2-order: (16)β(1+γ)(a12+4a0a2)λ2+β[-(γ+1)(γ-8)a1a2μ+3γ(1+γ)a0a1+(γ+1)(19γ-8)a2a-1]λ-2β(γ+1)(γ-2)a22μ2+2β(γ+1)(a12+4γa0a2)μ-vγ3(2a0a2+a12)+γ2[+a22a-2+2a1a2a-1)4β(a1a-1+4a2a-2)-v(a12+2a0a2)+3α(a0a12+a02a2+a22a-2+2a1a2a-1)]=0.

( 4 )   ϕ3-order: (17)βa1a2(γ+4)(γ+1)λ2+β(γ+1)×[(γ+2)a12+2(γ+4)μa22+10γa0a2]λ-2vγ3a1a2+[+α(a13+3a22a-1+6a0a1a2)2β(a0a1+5a2a-1)+2a1a2(βμ-v)+α(a13+3a22a-1+6a0a1a2)]γ2+2β[(3γ-2)a2a-1+(5γ+4)a1a2μ+γa0a1]=0.

( 5 )   ϕ4-order: (18)4(1+γ)βa22λ2+(γ+1)(5γ+8)βa1a2λ-vγ3a22+[3αa2(a12+a0a2)+β(a12+6a0a2)-va22]γ2+2βγ(a12+3a0a2+4a22μ)γ+8β(a22μ+a12)=0.

( 6 )   ϕ5-order: (19)γ2(4βa1+2βa2λ+3αa1a2)a2+2γβ(5λa2+4a1)+4β(2λa2+a1)a2=0.

( 7 )   ϕ6-order: (20)αγ2a23+2βγ(γ+1)(γ+2)a22=0. Equations for the coefficients of ϕ-i (i[1,6]) are similar to the above equations and hence not shown here.

It is straight forward to give the solution sets of the algebraic equations in different cases as follows.

Case i.

All the coefficients in (13) are equal to 0, and λ, μ, and v are arbitrary.

Case ii.

All the coefficients in (13) are equal to 0 except for a0=(1+γ)v/α, and λ, μ, v are arbitrary.

Case iii.

All the coefficients in (13) equal to 0 except for a2=-2β(γ2+3γ+2)/αγ2.

Case iv

Consider (21)a0=-2β(γ2+3γ+2)μαγ2,a-2=-2β(γ2+3γ+2)μ2αγ2,a1=a2=a-1=0,λ=0,v=-4βμγ2.

Case v

Consider (22)a1=-2β(γ2+3γ+2)λαγ2,a2=-2β(γ2+3γ+2)αγ2,a0=a-1=a-2=0,μ=0,v=βλ2γ2.

Case vi

Consider (23)a0=-2β(γ2+3γ+2)μαγ2,a1=-2β(γ2+3γ+2)λαγ2,a2=-2β(γ2+3γ+2)αγ2,a-1=a-2=0,v=β(λ2-4μ)γ2.

Case vii

Consider (24)a0=-β(γ2+3γ+2)λ22αγ2,a1=-2β(γ2+3γ+2)λαγ2,a2=-2β(γ2+3γ+2)αγ2,a-1=a-2=0,μ=λ24,v=0.

Case viii

Consider (25)a0=-4β(γ2+3γ+2)μαγ2,a2=-2β(γ2+3γ+2)αγ2,a-2=-2β(γ2+3γ+2)μ2αγ2,a1=a-1=0,λ=0,v=-16βμγ2.

Case ix

Consider (26)a0=-β(γ2+3γ+2)λ22αγ2,a-1=-β(γ2+3γ+2)λ32αγ2,a-2=-β(γ2+3γ+2)λ48αγ2,a1=a2=0,μ=λ24,v=0.

Case x

Consider (27)a0=-2β(γ2+3γ+2)μαγ2,a-1=-2β(γ2+3γ+2)λμαγ2,a-2=-2β(γ2+3γ+2)μ2αγ2,a1=a2=0,v=β(λ2-4μ)γ2.

Cases i to iii are trivial and of no interest, hence not discussed here. We focus our attention to cases from iv to x. Using solutions (21) to (27), solution (13) can be expressed in different forms corresponding to different cases listed above.

For Case iv, there are two solution types with ξ=x+4βμt/γ2.

(iv-1) When μ<0, we obtain the hyperbolic function solution: (28)w(ξ)=-2β(γ2+3γ+2)μαγ2+2β(γ2+3γ+2)μαγ2·(+2C1C2sinh-μξcosh-μξ-C12)-1((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C22((C12+C22)cosh2-μξ)×((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C12)-1), where C1 and C2 are integration constants. Recalling (11) we get (29)u(ξ)=[-2β(γ2+3γ+2)μαγ2+2β(γ2+3γ+2)μαγ2·(+2C1C2sinh-μξcosh-μξ-C12)-1((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C22)×((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C12)-1)-2β(γ2+3γ+2)μαγ2]1/γ. For simplicity, we only show expression for w rather than u in the following cases.

(iv-2) When μ>0, we have the trigonometric function solution: (30)w(ξ)=-2β(γ2+3γ+2)μαγ2+2β(γ2+3γ+2)μαγ2·(+2C1C2sinμξcosμξ-C12)-1((C12-C22)cos2μξ+2C1C2sinμξcosμξ+C22)×((C12-C22)cos2μξ+2C1C2sinμξcosμξ-C12)-1).

For Case v, there are also two types of solution with ξ=x-βλ2t/γ2.

(v-1) When λ0: (31)w(ξ)=-2β(γ2+3γ+2)λαγ2×(λ2C1sinh(λ/2)ξ+C2cosh(λ/2)ξC1cosh(λ/2)ξ+C2sinh(λ/2)ξ-λ2)-2β(γ2+3γ+2)αγ2×(λ2C1sinh(λ/2)ξ+C2cosh(λ/2)ξC1cosh(λ/2)ξ+C2sinh(λ/2)ξ-λ2)2.

(v-2) When λ=0: (32)w(ξ)=-2β(γ2+3γ+2)αγ2C22(C1+C2ξ)2.

For Case vi, there are three types of solution with ξ=x-β(λ2-4μ)t/γ2.

(vi-1) When λ2-4μ>0: (33)w(ξ)=-2β(γ2+3γ+2)μαγ2-2β(γ2+3γ+2)λαγ2×(+C2sinhλ2-4μ2ξ)-1)λ2-4μ2·(+C2sinhλ2-4μ2ξ)-1(C1sinhλ2-4μ2ξ+C2coshλ2-4μ2ξ)×(C1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ)-1)-λ2λ2-4μ2)-2β(γ2+3γ+2)αγ2·(+C2sinhλ2-4μ2ξ)-1)λ2-4μ2×(+C2sinhλ2-4μ2ξ)-1(C1sinhλ2-4μ2ξ+C2coshλ2-4μ2ξ)×(C1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ)-1)-λ2+C2sin4μ-λ22ξ)-1))2.

(vi-2) When λ2-4μ<0: (34)w(ξ)=-2β(γ2+3γ+2)μαγ2-2β(γ2+3γ+2)λαγ2×(+C2sin4μ-λ22ξ)-1)4μ-λ22·(+C2sin4μ-λ22ξ)-1(-C1sin4μ-λ22ξ+C2cos4μ-λ22ξ)×(C1cos4μ-λ22ξ+C2sin4μ-λ22ξ)-1)-λ2+C2sin4μ-λ22ξ)-1))-2β(γ2+3γ+2)αγ2·(+C2sin4μ-λ22ξ)-1)4μ-λ22×(+C2sin4μ-λ22ξ)-1(-C1sin4μ-λ22ξ+C2cos4μ-λ22ξ)×(C1cos4μ-λ22ξ+C2sin4μ-λ22ξ)-1)-λ2+C2sin4μ-λ22ξ)-1))2.

(vi-3) When λ2-4μ=0: (35)w(ξ)=-2β(γ2+3γ+2)μαγ2-2β(γ2+3γ+2)λαγ2(C2C1+C2ξ-λ2)-2β(γ2+3γ+2)αγ2(C2C1+C2ξ-λ2)2.

For Case vii, we only have one type of solution: (36)w(ξ)=-β(γ2+3γ+2)λ22αγ2-2β(γ2+3γ+2)λαγ2(C2C1+C2ξ-λ2)-2β(γ2+3γ+2)αγ2(C2C1+C2ξ-λ2)2, where ξ=x.

For Case viii, there are three types of solution with ξ=x+16βμt/γ2.

(viii-1) When μ<0: (37)w(ξ)=-4β(γ2+3γ+2)μαγ2+2β(γ2+3γ+2)μαγ2·(+2C1C2sinh-μξcosh-μξ-C22)-1((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C12)×((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C22)-1)+2β(γ2+3γ+2)μαγ2·(+2C1C2sinh-μξcosh-μξ-C12)-1((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C22)×((C12+C22)cosh2-μξ+2C1C2sinh-μξcosh-μξ-C12)-1).

(viii-2) When μ>0: (38)w(ξ)=-4β(γ2+3γ+2)μαγ2+2β(γ2+3γ+2)μαγ2·(+2C1C2sinμξcosμξ+C22)-1((C12-C22)cos2μξ+2C1C2sinμξcosμξ-C12)×((C12-C22)cos2μξ+2C1C2sinμξcosμξ+C22)-1)+2β(γ2+3γ+2)μαγ2·(+2C1C2sinμξcosμξ-C12)-1((C12-C22)cos2μξ+2C1C2sinμξcosμξ+C22)×((C12-C22)cos2μξ+2C1C2sinμξcosμξ-C12)-1).

(viii-3) When μ=0: (39)w(ξ)=-2β(γ2+3γ+2)αγ2(C2C1+C2ξ-λ2)2.

For Case ix, only one type of solution exits; it is (40)w(ξ)=-β(γ2+3γ+2)λ22αγ2+β(γ2+3γ+2)λ3αγ2C1+C2ξλ(C1+C2ξ)-2C2-β(γ2+3γ+2)λ42αγ2(C1+C2ξ)2[λ(C1+C2ξ)-2C2]2, where ξ=x.

For Case x, there are three types of solution with ξ=x-β(λ2-4μ)t/γ2.

(x-1) When λ2-4μ>0: (41)w(ξ)=-2β(γ2+3γ+2)μαγ2-4β(γ2+3γ+2)λμαγ2×(coshλ2-4μ2ξ)-1(C1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ)×(coshλ2-4μ2ξ(λ2-4μC1-λC2)×sinhλ2-4μ2ξ+(λ2-4μC2-λC1)×coshλ2-4μ2ξ)-1)-8β(γ2+3γ+2)μ2αγ2·(coshλ2-4μ2ξ]2)-1(C1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ)2×(coshλ2-4μ2ξ]2[(λ2-4μC1-λC2)sinhλ2-4μ2ξ+(λ2-4μC2-λC1)×coshλ2-4μ2ξ]2)-1).

(x-2) When λ2-4μ<0:(42)w(ξ)=-2β(γ2+3γ+2)μαγ2-4β(γ2+3γ+2)λμαγ2×(cos4μ-λ22ξ)-1(C1cos4μ-λ22ξ+C2sin4μ-λ22ξ)×(-(4μ-λ2C1+λC2)sin4μ-λ22ξ+(4μ-λ2C2-λC1)×cos4μ-λ22ξ)-1)-8β(γ2+3γ+2)μ2αγ2·(cos4μ-λ22ξ]2)-1(C1cos4μ-λ22ξ+C2sin4μ-λ22ξ)2×(cos4μ-λ22ξ]2[-(4μ-λ2C1-λC2)sin4μ-λ22ξ+(4μ-λ2C2-λC1)×cos4μ-λ22ξ]2)-1).

(x-3) When λ2-4μ=0: (43)w(ξ)=-2β(γ2+3γ+2)μαγ2+4β(γ2+3γ+2)λμαγ2C1+C2ξλ(C1+C2ξ)-2C2-8β(γ2+3γ+2)μ2αγ2(C1+C2ξ)2[λ(C1+C2ξ)-2C2]2.

As an example, we consider solutions of Case (iv-1) when C1=0 and μ=-1; then the solution reduces to the soliton form as (44)u(ξ)=12βαsech2(x-4βt) for γ=1 as the standard KdV case and (45)u(ξ)=6βαsech(x-βt) for the standard mKdV case when γ=2. We show the diagrams of Case (iv-1) in Figure 1 to illustrate the behaviors of the solutions for different power index γ. We choose coefficients of the generalized mKdV equation as α=6 and β=1; the constants of the solutions C1 and C2 equal 0.5 and 2, respectively, while the coefficient μ in the equation equals −0.2. It is found that (i) this kind of solution is a type of soliton solution; the amplitude decreases with the increase of power index γ; (ii) it gives that the width of the wave packet broadened when increasing γ; (iii) in addition, we find that the velocity of the wave slowed down when γ is bigger.

Figures of solutions of Case (iv-1). The solution u(ξ) evolutes with spatial coordinate x and time t. The subscript of u indicates the four kinds of different power index γ of the generalized mKdV equation.

4. Summary and Conclusion

In this paper, we use the modified (G/G)-expansion method to construct some types of solutions of the mKdV equation through the introduction of a proper transformation. Some new solutions are given, including the hyperbolic, trigonometric, and rational function solutions. It is shown that using the modified (G/G)-expansion method we can deal with the nonlinear evolution equations effectively and directly and abundant solutions could be obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge Qinghua Bu, Qingchun Zhou, and Mingxing Zhu for useful discussion. This work was supported by NSF-China under Grant nos. 11047101, 11105039, 11205071, and 11391240183.

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